Distribution of points on spheres and approximation by zonotopes (Q1117461)

In Euclidean space \({\mathbb{R}}^ n\) (n\(\geq 3)\) let \(B^ n\) be the unit ball and let P be a Minkowski sum of N segments. In a paper by the author and \textit{V. Milman} [Acta Math. 162, No.1/2, 73-141 (1989)] it was shown that \(B^ n\subset P\subset (1+\epsilon)B^ n\) implies \[ N \geq c_ 1(n)\epsilon^{-2(n-1)/(n+2)}. \] In the present paper it is proved that there is a constant \(c_ 2(n)\) so that for every \(0<\epsilon <1/2\) there is a sum P of N segments satisfying \(B^ n\subset P\subset (1+\epsilon)B^ n\) and \[ N \leq c_ 2(n)(\epsilon^{-2}| \log \epsilon |)^{(n-1)/(n+2)}. \] This improves a result of \textit{J. Linhart} [Arch. Math. 53, 82-86 (1989)]. Slightly weaker results are obtained for general zonoids instead of the unit ball \(B^ n\).